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Department of Economics

Financial Economics

Session 4: CAPM

Applicability, empirics and extensions

Postgraduate Class 2017

Economics Department Stellenbosch

Lecturer : Nico Katzke

[email protected]

News this week: 2014

• African Bank Investment Ltd (ABIL) had a rough old week…

• After the shock resignation of CEO Leon Kirkinis on

Wednesday (following an exceptionally poor trading statement

once again – a headline loss of over R6.4bn for the year to

September, requiring R8.5bn in new capital)– the share price

plummeted from R6.88 last Tuesday, to 30c by Friday…

shrinking the market value of ABIL from R10bn to R465mn in

one week!

• Although ABIL’s problems emerged last year with a report and

fine by the SA NCR.

2


• ABIL was yesterday placed under curatorship by the SARB.

• The SARB acted quickly and highly effectively in dealing with the

matter, as e.g. Kokkie Kooyman said:

• "I am impressed with the whole exercise and clarity of the Reserve Bank’s communication....

What they have done is bought time for the curator and managers to resolve outstanding

issues and eventually give shareholders a better basis on which to make a decision on

whether to put more capital on African Bank.

• The SARB split the Bank into a “Good bank” and a “Bad Bank” – buying the bad part

at a discount (paid R7bn for the R17bn part of the book – while the big four banks

and Investec and the PIC bought the “Good Part” of the book for R26bn.

• Trading of ABIL shares have been suspended on the JSE – and plans are being drawn

up to let it trade as the good bank part in time (to let investors opt in or out).

3


• Big News: China devalues its currency.

• This follows sustained strengthening of the dollar which has put strain on

the Chinese currency (Yuan) which largely tracks it.

• What does this imply?

• Could it signal structure weakness in the Chinese economy and that commodity

demand might be lower in the future?

• It might lead to the Fed postponing its rate hikes (as this would lead to a further

strengthening of the US$, and might lead to further relative Yuan depreciation –

which in turn could potentially hurt US trade competitiveness… Although this

VOX article suggests the US is in a better space today than it was in 2009, when

the Chinese were blamed for its competitive devaluation policies:

http://www.vox.com/2015/8/13/9149953/chinese-currency-devaluation-explained) 4

http://www.vox.com/2015/8/13/9149953/chinese-currency-devaluation-explained


• Chinese devaluation stirs our market:

• The JSE had a rough ride this week following two days of consecutive

devaluations by Chinese authorities.

• Resources bounced back somewhat, but Naspers in particular took shots

due to its large Chinese market exposure…

• The two sides of the devaluation:

• China’s imports are hurt – impact on commodity prices and the rand.

• US’ relative export competitiveness might lead the Fed not to raise rates in

September following better economic environment, as this could lead to further

Chinese devaluations…

5

News this week

• Tencent rally brings valuation of Naspers into sharp perspective…

• Its one third holding of Chinese tech giant, Tencent, which saw excellent

results boost the share price tremendously this past year – is worth

roughly R1.8trn.

• Yet Naspers currently has a Market Cap of roughly R1.27trn.

• THUS: You are getting paid well to hold all the other companies in the

Naspers stable.

• What’s going on here (and has been for years)?!

• Is Naspers (at close to R3000 per share) undervalued, or is the tech

industry (particularly the fickle Chinese market) – experiencing a bubble?

6

ANNOUNCEMENT: Essay!

• The essay will be due last Friday of September.

• The following applies:

• Topic to be finalized and communicated to me the week after the term

break

• Needs to be finance focussed

• Maximum of 2500 words.

• Marks awarded based on quality of writing, depth of research, innovative

thinking / strength of reasoning, how well you addressed and covered your

topic

• Turnitin submission on or before the final date.

• The topic needs to be interesting and informative – the idea being that you teach

yourself something that falls outside the scope of this course, but which interests

you.

• See it as an opportunity to spend time tooling up on an area in finance that you are

interested in

• Read the FT, Bloomberg News, VoxEU, finance blogs, etc. to get some idea of what

interests you!

7

Plan for today

• CAPM’s empirical performance.

• Growth vs Value assets

• CAPM’s use in practice: How its values are

determined and by whom

• Extensions of CAPM

• Empirical performance.

• Anomalies in finance

8

CAPM’s major contribution

• The asset pricing equation that we derived last session, CAPM, provided investors

with an important insight – in that the expected excess return on holding a

security is directly proportional to its systematic (market) risk, as given

by its 𝛽, and is not related to its idiosyncratic (asset-specific) risk.

• This may sound completely un-intuitive… but consider this:

• Such idiosyncratic risk is not compensated for, as such risk is assumed to be

completely eliminated in an efficient market by combining the asset with other

imperfectly correlated assets in a diversified portfolio.

• Outperforming the market, therefore, requires investors to take on higher than unity

𝛽 assets in theory – thereby accepting higher risk in hope of getting higher return

𝑹𝒌 = 𝑹𝑹𝑭 + 𝜷𝒌 . (𝑹𝑴 − 𝑹𝑹𝑭)

9

How are the parameters used calculated in

practice?

• Several firms in SA offer services that provide key measures to their

clients with which to calculate the relevant risk-adjusted returns of

financial assets and portfolios.

• The one we will focus on today is that of the BNP Paribas report

(available on the course web page)

10

Portfolio’s Beta

• For many institutional investors and firms alike, calculating the weighted 𝛽 of a

portfolio’s shares is an important indication of the amount of risk the portfolio is

exposed to (for fund managers, investors and regulators alike).

• This also allows investors to assess the relative performance of a portfolio.

• The relative performance of a portfolio is an important concept that uninformed

investors tend to ignore.

• In the past (and even still today) many investors evaluated the performance of a

portfolio based on its returns only.

• This is, however, a dangerous exercise as risk-unadjusted performance measures

ignore the risks taken to achieve the return.

• A portfolio that has a higher risk exposure should, by definition, offer investors

higher returns – otherwise they are not compensated for the potential downside

to the investment strategies employed.11

Portfolio’s Beta

• The premium paid (i.t.o. fees) to asset managers that attempt to outperform

the market → are (in theory) justified by their ability to correctly predict the

movement of the market (or individual assets).

• Shifting the weighted Beta of a portfolio is vital in correctly “timing” the market

(move to high beta stocks in bull market / low beta stocks in bear markets…)

• Thus, in theory – in a closed universe where investors face only a

choice of risky equities and the RF asset – if a portfolio A has

weighted 𝜷 = 𝟏. 𝟓 and portfolio B has weighted 𝜷 = 𝟎. 𝟖, we expect

portfolio A to outperform portfolio B on aggregate, due to the

larger risk undertaken (very broadly speaking)

• If A earns 7% and B 6%, how do we assess which portfolio delivered the best

relative performance?12

Portfolio’s Beta

Comparing the risk-adjusted returns of different portfolios by:

1 Divide annual return by the portfolio Beta.

This is known as the Treynor Ratio:

T =𝑅𝑃−𝑅𝐹

𝛽𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜

A higher Treynor ratio implies a portfolio having a higher risk-adjusted return.

The problem with this measure is that it implies that a portfolio is well diversified (so that 𝛽 =

good measure of the un-diversifiable risk the portfolio is exposed to)

2 We could also i.s.o. 𝛽 (which assumes a very well diversified portfolio) use the

Sharpe ratio

S =𝑅𝑃 − 𝑅𝐹𝜎𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜13

Portfolio’s Beta

• Notice that the Sharpe ratio measures performance relative to the CML, while the

Treynor ratio measures performance relative to the SML.

• Although T is easier to compute (easy to find weighted 𝛽), S is a better comparable

figure (although it assumes the risk is accurately reflected in past variation…)

• In our closed universe, a weighted portfolio 𝛽 of 0.7 should by definition yield the

same as a portfolio consisting of 70% M and 30% RF assets.

• Another measure widely used to compare portfolios is Jensen’s Alpha:

• Here we compare two portfolios that have the same 𝜷, and determine their 𝛼’s.

• Remember from the previous session that 𝜶 was an indication of excess return

(above expected). Thus if true return = 𝑅𝑃, then:

Jensen’s Measure: 𝛼𝑃 = (𝑅𝑃 − 𝑅𝐶𝐴𝑃𝑀) = 𝑅𝑃 − {𝑅𝐹 − 𝛽𝑃 𝑅𝑀 − 𝑅𝐹 }14

Portfolio’s Beta

• Now suppose a portfolio manager earned a high 𝜶 in the last year.

• This implies that the return on his portfolio was higher than expected (or

above-average considering the risk-objectives)

• Is this in any way an indication of future positive and high 𝜶?

• Not even close.

• Perhaps it was luck. Or perhaps the manager hid some of the risk the

portfolio was exposed to (remember 2008 – many portfolio managers

scampered to stock up their funds with high earning “Super Safe” MBS

instruments… clearly belying their true portfolio risk and creating the

illusion of superior 𝛼′𝑠).

• Or perhaps (and I’m sure the manager will agree…), it was just down

to unique skill…15

What it then boils down to…

• Active investment strategies then boils down to chasing alphas.

• This chase can, however, yield an unfruitful exercise on aggregate → as the

costs erode the return on aggregate.

• The problem then is – in the short run your manager might have a high 𝛼, but

over the long term will probably return the market return – minus costs, it

would leave you in a negative sum game…

• See Bogel’s cynical, yet insightful paper on investments, as well as Nassim

Taleb’s interesting insight on the difference between noise and

information - posted on the site… is chasing alpha based on noisy

patterns a wise thing? 16

How are these measures computed in

practice?

• As mentioned, we will now look at how industry calculates the parameters

that we can use to evaluate assets and portfolios.

• First, the monthly (or whichever you have available) returns are calculated

for each of the relevant shares on the JSE, while the corresponding returns

of the JSE/FTSE ALSi (All Share Index) are also calculated.

• They then regress the two series as follows (For Anglo Am & JSE ALSi):

17

Source:

Cadiz:

Equity Risk Service

on the JSE


practice?

• The individual shares and separate indexes (including, among many, the JSE Top 40-,

Financial-, Venture Capital-, AltX-, Listed Property-, Value- & Growth Indexes and also

all the shares listed) are then regressed relative to the Benchmark

• The Benchmarks in the report are: the JSE/FTSE ALSi, Financial and Industrial Index, Resource

top 20 Index & the JSE Top 40 index.

• Several key indicators are then provided for stock / indexes relative to the Benchmark

used:

1. 𝛃 → the regression line fitted using the regression techniques: This is (as hammered home

by now) an indication of the sensitivity of a share / index’s price change relative to the

benchmark (market).

2. Annualised 𝜶 → Avg return if the Benchmark had a zero change (thus the intercept of the

regression line): proxying excess returns for holding this share…

18


practice?

3. SE(𝛃) → This is an indication off the reliability of the estimate of β. This is used to construct a

C.I. that we assume with 95% assurance contains the true β. By definition, a low SE indicates an

accurate estimate of β.

4. Annualized Total risk = 𝜎Returns : This measures the share / index’s total risk in % (but

remember our contention about risk measured as % deviation of returns…). To make the

monthly adjustment – divide the % by 12.

5. Annualized Unique risk = Fluctuations due to events unique to the company (strikes, bad

management, law suits…)

6. 𝐑𝟐 → Proportion of total risk accounted for by the market (systematic) risk.

A high 𝑹𝟐 implies much of the variation in Assets returns are caused by the Market’s

variation. Low 𝑹𝟐 → large proportion of movements due to unique factors

19

A few such examples of

Indexes relative to JSE ALSi (Dec 2014)

20

Expected total

monthly

variation:

72.4%/12 =6%

Index 𝜷 SE(𝜷) Ann Total

Risk (𝝈%)

Ann

Unique

Risk (%)

𝑹𝟐

JSE ALSi 1.00 0.00 18.7 0 100%

Top 40 1.06 0.01 18.7 1.7 99%

Resource

top 20

1.3 0.08 26.7 11 83%

Industrial

top 25

0.78 0.07 17.3 9.2 71%

Financial

top 15

0.74 0.11 21.1 15.8 43%

Venture

Capital

Index

1.5 0.49 72.4 66.2 15%

Alt X 0.61 0.17 26.1 23.3 19%

Value

Index

0.94 0.04 18.3 5.1 93%

Growth

Index

1.07 0.04 20.8 5.4 92%

A few such examples of

Shares relative to JSE ALSi (Dec 2014)

21

Share 𝜷 SE(𝜷) Ann Total

Risk (𝝈%)

Ann

Unique

Risk (%)

𝑹𝟐

ABSA 0.56 0.16 24.5 22 18%

FNB 0.78 0.18 28.9 24.7 25%

Nedbank 0.65 0.16 25.9 22.7 22%

Standard

Bank

0.74 0.17 28 24.1 25%

Capitec 0.63 0.2 30.2 27.6 15%

Old

Mutual

1.18 0.22 33.6 25.1 43%

Sanlam 0.81 0.14 25.1 19.8 37%

Anglo Am 1.67 0.16 37.8 21.2 68%

Shoprite 0.38 0.17 24.1 22.8 8%

MTN 0.84 0.16 27.1 22 34%

Vodacom 0.3 0.22 17.9 17.1 6%

Sasol 1.05 0.11 24.6 14.8 63%

Considerations when using such quantitative

indicators

• For shares that are thinly traded, the estimate of 𝛽 will be less accurate – the

Cadiz report controls for this.

• Time horizon → the report uses 5 years of monthly data (60 data points),

which might be subject to small sample biases – but the trade-off is that a

monthly frequency is subject to less noise and errors than higher frequency

(weekly / daily).

• Stability of Betas → as mentioned before, Betas can and do change over time.

This can be problematic for investors hoping to “cash in” on a high beta asset in

a bull market, only to find a dampened upside and an intensified downside made

the Beta look large…

• (for more details on the practitioners guide to calculating this measure… See:

EstBeta.pdf on webpage)

22

Considerations when using such quantitative

indicators

• As these indicators are based on past co-movements of the shares and the market, several factors

may influence its accuracy in determining future co-movements:

• Accuracy of past data – can be questioned as the past five years include a heavily distorting

global financial crisis.

• Intensified interconnectedness of certain shares in your portfolio at times, that negate the

diversification benefits and causes the portfolio to vary less than in accordance to the market.

• Unexpected events / unique factors of a business might become more pronounced in the

future – thus investors should be aware of this and keep it in consideration. This emphasizes

the need for qualitative evaluations and forecasted quantitative evaluations too.

• Some businesses listed on the JSE might be more influenced by external factors /

macroeconomic factors that do not necessarily influence the market. This can be seen by

viewing the low 𝑅2, which may not necessarily be a good indication of this.

23

Market Beta = Naspers?!

• Today the Beta measure has often been criticized

as implying the comovement largely with Naspers

(and a few other stocks).

• Which, indirectly, implies the comovement with the

Chinese market via tencent.

• A couple of years ago Beta was tilted towards the

resource sector (and thus showed high

comovement with commodity prices).

24

Examples of how we can use the indicators.

• Suppose you were provided with the tables above as given.

• Calculate the Beta and Risk of a portfolio that consists of the following asset

distribution (the nice thing about finding the 𝛽𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 is that it can simply be

added with the respective weightings):

25

SHARE % of portfolio

ABSA 30

ANGLO AMERICAN 20

MTN 10

JSE ALSi 30

R157 Bond 10

Portfolio Beta : σ(Weight) . (𝛽𝑆ℎ𝑎𝑟𝑒𝑠)

𝛽𝑃 = 0.3 0.56 + 0.2 1.67 + 0.1 0.84 + 0.3 1 + 0.1 0 = 0.886

Examples of how we can use the indicators.

Calculate the total Portfolio Risk:

Total Risk = Systematic (market) Risk + Non-Systematic (unique) Risk

𝑇𝑜𝑡𝑎𝑙 𝑅𝑖𝑠𝑘2 = 𝑀𝑎𝑟𝑘𝑒𝑡 𝑅𝑖𝑠𝑘2 + 𝑈𝑛𝑖𝑞𝑢𝑒 𝑅𝑖𝑠𝑘2

𝑇𝑜𝑡𝑎𝑙 𝑅𝑖𝑠𝑘2 = 𝑅2 𝑇𝑜𝑡𝑎𝑙 𝑅𝑖𝑠𝑘 2 + (1 − 𝑅2) 𝑇𝑜𝑡𝑎𝑙 𝑅𝑖𝑠𝑘 2

∴ Market risk of this proposed portfolio:

Market risk = 𝛽𝑃 . 𝑇𝑜𝑡𝑎𝑙 𝑟𝑖𝑠𝑘 𝑜𝑓 𝑡ℎ𝑒 𝑀𝑎𝑟𝑘𝑒𝑡 𝑖𝑛𝑑𝑒𝑥 = 𝛽𝑃 . (𝜎𝐽𝑆𝐸_𝐴𝐿𝑆𝑖)

= 0.886 18.7 = 𝟏𝟔. 𝟓𝟔%

Unique Risk of the portfolio: (bit more complicated)

Unique Risk = σ(𝑊𝑒𝑖𝑔ℎ𝑡). ( 1 − 𝑅2 ∗ 𝐴𝑛𝑛 𝑇𝑜𝑡𝑎𝑙 𝑅𝑖𝑠𝑘 2])

= 0.3 ∗ [ 1 − 0.18 . 24.5 ]2 + 0.2 * [ 1 − 0.68 . 37.8 ]2

+0.1* [(1 − 0.34) 27.1 ]2 = 𝟏𝟑. 𝟐%

26

Calculating the total Portfolio Risk:

27

Total Risk = Systematic (market) Risk + Non-Systematic (unique) Risk

𝑇𝑜𝑡𝑎𝑙 𝑅𝑖𝑠𝑘2 = 𝑀𝑎𝑟𝑘𝑒𝑡 𝑅𝑖𝑠𝑘2 + 𝑈𝑛𝑖𝑞𝑢𝑒 𝑅𝑖𝑠𝑘2

= 16.562 + 13.22 = 448.4

Thus the Total Risk of the portfolio that we proposed would be:

448.4 = 𝟐𝟏. 𝟏𝟕% 𝑷.𝑨

And the portfolio’s 𝑅2 =𝑀𝑎𝑟𝑘𝑒𝑡 𝑟𝑖𝑠𝑘

𝑇𝑜𝑡𝑎𝑙 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑅𝑖𝑠𝑘=

16.562

448.4= 0.61

& ∴ 1 − 𝑅2 = 0.39

So that the movement in the JSE ALSi explains approx. 𝟔𝟏% of the movement in our portfolio

– implying great scope left to diversify portfolio further & make the 1 − 𝑅2 statistic less (and

thus be less exposed to unique risks!!)

Abnormal Portfolio return

• Suppose you are given the following returns information at the end of the year.

• Calculate the portfolio’s abnormal return:

Remember: 𝛽𝑃 = 0.886

This implies the benchmark risk-adjusted return (i.e. of a portfolio having 88.6%

invested in JSE ALSi and 11.4% invested in R157 Bond) is:

0.886 8% + 1 − 0.886 5.5% = 𝟕. 𝟕𝟏𝟓%

28

SHARE Return

ABSA 15 %

ANGLO AMERICAN -3 %

MTN 10 %

JSE ALSi (𝑅𝑀) 8 %

R157 Bond (𝑅𝑅𝐹) 5.5 %


• Our portfolio return would then be:

0.3 15% + 0.2 −3% + 0.1 10% + 0.3 8% + 0.1 5.5% = 𝟕. 𝟖𝟓%

The Abnormal Return of our portfolio is thus:

7.85% − 7.715% = 0.135%

29

SHARE Portfolio Weight Return

ABSA 30% 15 %

ANGLO

AMERICAN

20% -3 %

MTN 10 % 10 %

JSE ALSi 30% 8 %

R157 Bond 10% 5.5 %


• Before applauding the portfolio manager too much, perhaps this

abnormal return of 𝟎. 𝟏𝟑𝟓% needs to be put into perspective…

• Although the Market risk of our Portfolio and the Benchmark portfolio

(on the SML in the CAPM) is identical, our portfolio has more unique

risk (𝟏𝟑. 𝟐% compared to the 𝟎% of the benchmark portfolio having

88.6% in M and 11.4% in RF).

• This implies that investors are merely compensated (with 0.135%) for

holding more unique risk in a less diversified portfolio…

• This follows as there is no compensation for holding unique risk

→ the core of the CAPM insights!!

30


• We can now calculate the two measures mentioned earlier:

T =𝑅𝑃 − 𝑅𝐹𝛽𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜

=7.85 − 5.5

0.886= 2.65

S =𝑅𝑃 − 𝑅𝐹𝜎𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜

=7.85 − 5.5

21.17= 0.11

• As mentioned earlier, however, due to the portfolio not being

highly diversified – it is better to use the Sharpe ratio in this

case (as the 𝛽 used in T assumes a sufficiently diversified portfolio).

31


CAPM’s Relevance today:

Survey of SA businesses

Correia and Cramer (2008)

CAPM’s relevance in SA

• CAPM, most likely due to its simplicity and applicability – is still

very much a useful starting point for any quantitative analysis of a

financial asset’s value, and forms the basis for more advanced

techniques.

• In fact, according to Correia and Cramer (2008: 41) the vast

majority of companies in SA use CAPM (or some basic derivative

thereof) to determine the fair cost of equity.

• The extensions thereof, such as APT and the dividend discount model,

are almost never used in the valuation of equity in practice

• The authors indicate a great movement toward explicitly valuing

equity i.t.o. CAPM (both in SA and abroad) over the last 15 years.

33

CAPM’s relevance in SA: RF

• What do firms use as the Risk Free rate?

• Correia and Cramer found the choice of bond yields to use in acting as a

proxy for the RF is quite varied. Although McKinsey & Co. recommend the

use of longer term bond yields that are less volatile, firms used a great

variation of bond yields (such as the R157 (maturing 2015), R201 (2014),

R186 (2026)*,,…) as a proxy for the RF. In fact, SA firms have shown to be

more inclined to use the shorter term yield (R157).

• Clearly there are no clear consensus of which bond yield to use as RF

proxy (which of course changes the pricing of equity between firms…

necessarily a bad thing for market liquidity?)

• *These are names of bonds with different maturities*

• We will be covering it in more detail in a later session, but for now you can check

out table 11.4 in the textbook for a more thorough indication.

34

CAPM’s relevance in SA: Equity premium

• The equity market premium of the CAPM: (𝑹𝑴 − 𝑹𝑹𝑭), reflects the

additional return that investors require for investing in the market

portfolio (or for holding equity in general).

• As with the proxy for RF, there is no clear consensus as to the calculation

of the equity premium for SA.

• Firms in SA have shown to use an equity premium of roughly 5.35%, while

the median is 5%.

• This is more or less consistent with the literature, with Kruger (2005)

estimating it at 5 – 5.5% and Dimson, et al (2003) at 5.2% historically.

35

CAPM’s relevance in SA: 𝛽

• Calculating accurate Betas of firms in SA seems an onerous task, considering all the

pitfalls and caveats..

• Lack of liquidity (of smaller firms whose assets do not trade as often as larger firms), arbitrary

time horizons and random occurrences that may not repeat in the future all cause

inaccuracies of the calculations of 𝛽.

• As seen earlier, firms can use the services of Cadiz / Bloomberg / McGregor / various other

sources – to calculate SA firm Betas .

• The most popular are Cadiz (25%), Bloomberg (25%) and McGregor (20%), while less

than 5% make in-house calculations].

• Correia & Cramer (2007)’s survey does, however, suggest that 44% of companies do

make some adjustments to their supplied Betas.

• The market portfolio (M) that is most widely used is the JSE ALSi (77%), but some firms see

this index as over-weighted toward large institutions (that assert strong individual influence on

the market outcome) and rather use the Financial and Industrial Index (FINDI) (roughly 23%) 36

CAPM’s relevance in SA

• Correia & Cramer (2008)’s survey therefore indicate that

SA firms “mostly adhere to practices that are consistent

with finance theory” – which reflects a highly developed

corporate and financial sector comparable with the US.

• They also suggest widespread use of key financial ratios in

determining the value of firms. This is important in order to

provide further quantitative assessments of the fundamental

valuations of firms.

• These ratios, and other important aspects, will be discussed

in two sessions’ time.37


Variations of CAPM

Backward looking syndrome

• As became clear again following the recent financial turmoil, assuming

normality of returns in stock markets and implicitly assuming that what

happened in the past is a good indication of what will happen in the

future are two of the most pertinent problems with modern financial

modelling.

• The CAPM (and many subsequent models of which it’s theory is at the

core), does not describe or predict the source of risk (i.e. it does not

specify what causes the risk premiums), but merely predicts its

correlation with the market return, and then provides from it a forecast

basis for future risk premiums.39

Introducing the second APM:

Arbitrage Pricing Theory (APT)

• The APT approach seeks to remedy this problem by specifying various

independent macro-economic variables which predict certain sources of

risk (and hence deducing the fair price from it).

• APT – Ross (1976):

• Uses linear Factor Analysis to identify common factors in the

risk of security returns

• These factors are used to explain the unsystematic (or

idiosyncratic) risk of securities

• Returns do not have to be assumed jointly normally distributed

as this is achieved through the linearity of the factor analysis

(addresses some shortcoming of the CAPM)

40

APT – underlying assumptions

• It assumes that returns can be described in terms of a linear factor model.

• In this model, the return on any security, k, is dependent on many factors, not just the

market ( having in total 𝐺 independent factors that influence the random return of

asset 𝑘) :

41

(g)factor of (risk)y volatilit theto

return s)'(security theofy sensitivit the ][ :With

,~~...

~~

,

,11,

k

kffRR

gk

kGGkkMk

෨𝑅𝑘 → Expected return of (k)

𝑅𝑀 → Expected market return෩𝑓1 → deviation of factor 1 from its expected (mean)value

→ ෩𝑓1 > 0 is thus a shock that is not priced in (𝐸(෩𝑓1) = 0)

ǁ𝜀 → Noise, which we assume can be diversified away!

APT further….

• To simplify we use (the concept of) factor-specific ‘mimicking portfolios’

(i.e. create portfolios which mimic the key factors)

• Hence a portfolio is created which is perfectly correlated with the specific factor, but

has zero correlation with all the other factors in the model.

• This is because the model requires independent factors as causal variables

• The mimicking portfolio then has a sensitivity, 𝛽𝑔, of (close to) 1 to the specific factor

[𝑔], and (close to) 0 to the other factors in the model

• We view the deviations as the returns to these portfolios…

• NB: this is a functional, not a causal relationship

• The factors (or mimicking portfolios) are only correlated with the measured asset 𝑘’s

returns – they don’t necessarily cause the returns

• They are usually macro-type factors42

][ ggg FFf ~~

Example of APT: Macro-Factors

• An example of this could be: if asset (𝑘) is an Anglo American (gold

mining company) share, while factors influencing it could be:

• 𝑔 = (𝐺𝑜𝑙𝑑 𝑃𝑟𝑖𝑐𝑒, 𝑅/$ 𝐸𝑥𝑐ℎ𝑎𝑛𝑔𝑒, 𝐺𝐷𝑃,… )

• Notice : That each of these factors are not necessarily drivers (Like

company profit / …) of the share value directly, but merely

correlated with the value of the shares.

• They, in a sense, provide a proxy for potential deviations from

the expected returns of holding the Anglo American share.

• [Note too : Gold Price In this case captures only the deviation

from the expected mean of the price of gold, not its level

changes.]43

Example of APT: Smart Beta

• The advent of Smart Beta products imply investors define certain

characteristics of companies in a systematic way.

• This implies adding additional Betas to the CAPM equation in a structured,

linear way.

• E.g., suppose investors want to invest in the highest Quality (ROE, ROA,

ROIC, etc), best Momentum (say 120day MovingAvg), most stable (lowest

120 SD) best Value (EbitDA / EV) stocks, one could assign scores for each

stock based on these factors, and create a portfolio based on such factors:

• 𝑹𝒌 = 𝑹𝑹𝑭 + 𝜷𝒌 . 𝑹𝑴 − 𝑹𝑹𝑭 + 𝜷𝒌𝑸 + 𝜷𝒌𝑽𝒂𝒍𝒖𝒆 +

𝜷𝒌𝑴𝒐𝒎𝒆𝒏𝒕𝒖𝒎 + 𝜷𝒌 𝑽𝒐𝒍

44

APT

• We will return to APT as an investment vehicle after our

discussions of financial instruments in the next few sessions.

• This will enable us to throw around some of the lingo to get

an understanding of how we can apply some of the insights

provided in the APT framework to designing an actual

investment vehicle:

• not by using immeasurable / abstract factors, but by

specifying actual factor proxies and thereby designing

portfolio weights.

45

Other variants of the CAPM: ICAPM

• The traditional CAPM (when used in a multi-period context) assumes:

• Interest rates and relative commodity prices are fixed (In order for two

period assumption to hold)

• Share returns are i.i.d. Normally distributed

• It isolates consumption risk by the single state variable: the variance in return

• Merton (1973a) extends the CAPM to a continuous time framework

known as the ICAPM.

• It assumes that: Changes in the aggregate pricing risk can be described

by the interest rate → which is assumed the only variable required

to represent changes in the investment opportunity set over time.

46

ICAPM

• Intuitively the ICAPM takes account of the covariance of the asset return

with the market return (as earlier implied by the CAPM, being regarded

as compensation for holding non-diversifiable risk due to changes in the

market)

• BUT also of the asset and the market interest rate (proxied for by the

derivative portfolio [z]).

• The rate of interest can be though of as compensation for the non-

diversifiable changes in consumption risk over time … i.e. changes in

the consumption- or investment opportunity set of consumers that change

over time)

47

CCAPM

• All the variations of the standard portfolio theory done so far

assume that all income is generated from the same source (that of

the investment portfolio).

• CCAPM seeks to remedy this by using instead of the Market

portfolio a proxy for aggregate consumption capabilities of

investors.

• The model assumes that income (in whichever capacity earned) is a

good indication of consumption ability… and therefore the argument

is made to study a portfolio’s returns relative to investors’

consumption abilities (as mimicked by an aggregate income portfolio).

48

Comparison of the variations of MPT pricing

theory models

Model Strengths Weaknesses

CAPM • Single factor identified in the

model

• Simple to use and data easily

available

• Various (mentioned earlier)

ICAPM • Dynamic version of (static)

CAPM, i.e. CAPM through

time

• Nature of factors not specified

in the model

• No other sources of income

CCAPM • Single factor identified in the

model

• Difficult to measure changes in

aggregate consumption

accurately

APT • Not forced to use single factor

• Potentially more accurate

• Nature of factors not specified

in the model.

• This is then especially open to

data-mining problems

49

Alternatives to MPT

• Modern Portfolio Theory and the subsequent asset pricing

models derived from the MVF framework have some serious

shortcomings.

• Yet it is still widely used in practice, despite several other frameworks that

have emerged as more accurate representations of investor behaviour.

• One such framework is the PMPT (Post MPT) framework that

optimizes a portfolio based on returns vs downside risk.

• Risk as defined by variance is a poor proxy for the human

perception of risk – which is the fear of an inferior outcome to

that which is expected (i.e. downside failure).

50

Alternatives to MPT

• Despite the clear benefits to more improved theoretic foundations

possible and better descriptions of risk, the MVF framework is still

the most widely used means of describing investor preferences.

• The question is then why the foundation of finance remains these

simple definitions and are used to guide portfolio decisions.

• Swisher & Kasten (2005) asks this question, and compares it to the

reluctance of the erstwhile medical industry in the 1900s to adopt

the use of sterilization – implying there is always reluctance to

adopt better paradigms, even when we know full well they are

superior...

51


Empirics

Empirical performance of Asset Pricing Models

• In addition to some of the shortcomings mentioned in the

previous session regarding CAPM’s stringent assumptions –

several other problems emerged in the literature that

questioned the validity of the models.

• A large body of literature emerged that tested the use of these

models in providing quantitative assessments of the fair price

of equity in the market – and showed that at times certain

inconsistencies and anomalies emerge.

• Some of the problematic assumptions that were subsequently

exposed (in addition to those mentioned earlier) include:

53

How robust are the assumptions? What are

the implications of relaxing them?

Assumption Validity? Implications if we relax the

assumption

Investors are risk averse Very robust The problem is not so much that

investors do not display some aversion to

risk, rather that they can at times and for a

portion of their portfolio welcome risk in

an effort to gain higher returns.

Future consumption is

funded solely by future

returns from portfolios of

securities

Not valid CCAPM developed in attempt to

correct for this assumption

Returns are joint normally

distributed

Not valid –

fat tails exist

(Black Swan

events can &

do happen)

Can’t describe returns in terms of mean

and variance only!

Market returns can and do at times

behave very differently to what its

historical record shows.

54



Assumption Validity? Implications?

Homogenous

expectations

Unlikely due to

costly information

No single risk pricing model can exist –

because all agents have unique pricing

models based on their information sets

(but we care more about the subjective

Efficient Frontiers and CML & SML!)

No constraints on

short

sales/borrowing

Not valid for all

markets/securities

CML becomes non-linear above

market portfolio – can only invest in

risky assets beyond this point

Arbitrage may be limited = presence

of abnormal returns which is

inconsistent with model’s pricing

equation

Presence of a risk-

free security

Not perfectly valid

(We listed 3 reasons

earlier…)

Can adapt the model by “creating” a

RF asset through bundling e.g.

Black’s (1972) zero Beta portfolio

creation.

55



Assumption Validity? Implications?

Taxes and

transaction costs

Not valid Can cause non-unique results, but

only if there are different marginal

tax rates and opportunity costs –

usually ignored in practice…

Two time periods Only if interest rates

and relative prices stay

constant and security

returns are IID (Thus

independent of state) –

very unlikely!

Merton’s (1972) Intertemporal

CAPM (ICAPM) developed to

remedy this problem...

Type of asset does

not matter

Not valid:

Various studies found

certain anomalies (e.g.

size, type, value of

assets do in fact

matter!)

Assets perform different to what

the CAPM predicts after

controlling for Beta – extensions to

the models made to account for

these anomalies (e.g. including

firm size / other indicators )

56

Early empirical tests

• Early empirical tests studied the ability of the models (esp. the CAPM) to explain the risk in

security returns in practice

• Black et al. (1972) sorted NYSE shares from 1931 - 1965 into 10 decile portfolios and

estimated the coefficients on the CAPM equation for these portfolios. They then tested the

following simple hypothesis:

• They found that:

• 𝝀𝟎 > 𝟎 and 𝝀𝟏< (𝒊𝑴 – 𝒊) → ∴ shares with low (high) Betas pay higher (lower) returns

than predicted by the CAPM

• It also found that, compared to other more advanced derivatives of the CAPM, the

simple version’s 𝛽 is still the best performing risk measure.57

holds CAPM theIf: ][ and ]0[

: thathypothesis With the

][

00

10

RFm

kkRFTrue

RR

RR

Early empirical tests continued

• Roll’s (1977) critique:

• As mentioned before, the tests of the CAPM are based on an

assumption that the market portfolio is correctly specified ex ante

• All subsequent empirical tests of the CAPM are therefore joint

tests on the: CAPM & M

• BUT specifying the latter is impossible in practice (i.e. using history) –

so no tests are (or ever can be) valid!

58

Further empirical tests

• Fama and French (1993) also showed that a risk model which includes

measures such as firm size & book to market equity ratios, in conjunction

with Beta, provided the best explanation of the cross-sectional variability of

security returns

• Fama and French, e.g., showed that firms that are small and firms with low P/E

ratios tend to out-perform the market on a risk-adjusted basis, concluding that in

many cases outperformance by assets is uncorrelated with their relative

sensitivity to market movements: 𝛽.

• F&F also suggested that Value stocks (Low P/Book or P/E ratio) tend to

outperform Growth stock (High Expected Earnings, thus often having

high P/E’s) irrespective of the Betas.59

Further empirical tests

• These F&F findings have in subsequent years been repeated across various

time-frames and various markets. It seems to clearly suggest that more

than one systematic risk (in addition to the market risk) impacts the price

of assets, in addition to relative market risk.

• Although F&F’s work did have some criticism – notably that size factors do

not explicitly relate to risk (else small firms would merely unite with

other small firms and thus reduce the size premium), while value factors is

based on equal weighting of small and large firms…. –

• Their later work still defiantly show that: P/E, Size, Debt/Equity,

Price/Book add to the explanation of expected stock returns given by

𝛽… Which is kind of stating the obvious…

60

Pricing Puzzles

• Mehra and Prescott (1985) approached the testing problem in a

fundamentally different way:

• They looked at the relationship between the values of the Constant relative

risk aversion measure (CRRA) and the (constant) rate of time preference

that’s implied in the observed patterns of security returns, and whether they

were consistent with those identified in experimental evidence (Thus are

people as risk averse as the models imply?)

• Well.. they weren’t (not by a long way)!! –Two puzzles emerged:

61

Pricing Puzzles

• The equity premium puzzle – investors require a far larger

premium for holding shares, as compared to bonds, than expected in

theory (CRRA needs to be at least 5 times larger than experimental evidence

suggests it should be)

• The risk free rate puzzle – the observed RF rate is much

lower than the SML would predict (for the minimum req’d rate), or the

CCAPM would predict for a realistic CRRA, and even more so when the

required CRRA (to solve the equity risk premium puzzle) is used.

• Hasan and van Biljon (2009) also study these puzzling inconsistencies

from the SA perspective and find that it delivers similar

inconsistencies.62

Summary

• Despite exhibiting at times poor empirical results (although the

CAPM’s performance have shown some of the best results, even

against far more advanced, technical and intricate pricing models),

the standard MPT theoretic framework and the subsequent

CAPM asset pricing model – remains at the core of finance today.

• Used widely and successfully in practice, one of its major benefits

is its simplicity of calculation and ease of interpretation.

• It provides investors and investment intermediaries with a good

quantitative basis from which to evaluate asset prices.

• Its drawbacks and limitations are clear, yet does not invalidate its

use.

Just… use it with care!63


END!

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